1st Edition
By: Carlos David Montilla Colmenarez 1
Fluid mechanics is the branch of physics that s t u dies fluids (liquids, gases , andplasmas) and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms, that is, it models matter from a macroscopic viewpoint rather than from a microscopic viewpoint.
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Fluid mechanics, especially fluid dynamics, is an active field of research with many unsolved or partly solved problems. Fluid mechanics can be mathematically complex, best be solved by numerical methods, typically using computers. A modern discipline, called computational fluid dynamics (CFD), is devoted to this approach to solving fluid mechanics problems. Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.
The study of fluid mechanics goes back at least to the days of ancient G r e e c e , when Archimedes investigated fluid statics andbuoyancy and formulated his famous law known now as the Archimedes' principle, which was published in his work On Floating Bodies- generally considered to be the first major work on fluid mechanics. Rapid advancement in fluid mechanics began with Leonardo da Vinci (o bserva tio ns a nd e xperime n ts ), E va n g e lis ta To r r ic e lli (invented the barometer), Isaac Newton (investigated viscosity) a n d B l a i s e P a s cal (researched hydrostatics, formulated Pascal's law), and was continued
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Inviscid flow was further analyzed b y va r io us ma t he ma t ic ia ns (Leonhard Euler, Jean le Rond d'Alembert, Joseph Louis Lagrange, Pierre-Simon Laplace, Siméon Denis Poisson) and viscous flow was explored by a multitude of engineers including Jean Louis Marie Poiseuille and Gotthilf Hagen. Further mathematical justification was provided by ClaudeLouis Navier and George Gabriel Stokes in theNavier–Stokes equations, and boundary layers were investigated (Ludwig Prandtl, Theodore von Kármán), while various scientists such as Osborne Reynolds, Andrey Kolmogorov, and Geoffrey Ingram Tay-
Like any mathematical model of the real world, fluid mechanics makes some basic assumptions about the materials being studied. These assumptions are turned into equations that must be satisfied if the assumptions are to be held true.
For example, consider a fluid in three dimensions. The assumption that mass is conserved means that for any fixed control volume (for example a sphere) – enclosed by a control surface – the rate of change of the mass contained is equal to the rate at wh i c h m a s s i s p a s s i n g from outside to inside through the surface, minus the rate at which mass is passing the other
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Fluid mechanics assumes that every fluid obeys the following: Conservation of mass Conservation of energy Conservation of momentum The continuum hypothesis, detailed below. Further, it is often useful (at subsonic conditions) to assume a fluid is incompressible – that is, the density of the fluid does not change. Similarly, it can sometimes be assumed that the viscosity of the fluid is zero (the fluid is inviscid). Gases can often be assumed to be inviscid. If a fluid is viscous, and its flow contained in some way (e.g. in a pipe), then the flow at the boundary must have zero velocity. For a viscous fluid, if the boundary is not porous, the shear forces between
Fluids are composed of molecules that collide with one another and solid objects. The continuum assumption, however, considers fluids to be continuous. That is, properties such as density, pressure, temperature, and velocity are taken to be well-defined at "infinitely" small points, defining a REV (Reference Element of Volume), at the geometric order of the distance between two adjacent molecules of fluid. Properties are assumed to vary continuously from one point to another, and are averaged values in the REV. The fact that the fluid is made up of discrete molecules is ignored. 5
The continuum hypothesis is basically an approximation, in the same way planets are approximated by point particles when dealing with celestial mechanics, and therefore results in approximate solutions. Consequently, assumption of the continuum hypothesis can lead to results which are not of desired accuracy. That said, under the right circumstances, the continuum hypothesis produces extremely accurate results.
The Navier–Stokes equations (named after ClaudeLouis Navier and George Gabriel Stokes) are the set of equations that describe the motion of fluid substances such as liquids and gases. These equations state that c h a n g e s i n momentum (force) of fluid particles depend only on the external pressure and internal viscous forces (similar to friction) acting on the fluid. Thus, the Navier– Stokes equations describe the balance of forces acting at any given region of the fluid.
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The Navier–Stokes equations are differential equations which describe the motion of a fluid. Such equations establish relations among the rates of change of the variables of interest. For example, the Navier–Stokes equations for an ideal fluid with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure.